Hubbard and Heisenberg models on hyperbolic lattices: Metal-insulator transitions, global antiferromagnetism, and enhanced boundary fluctuations

Abstract

We study the Hubbard and Heisenberg models on hyperbolic lattices with open boundary conditions by means of mean-field approximations, spin-wave theory, and quantum Monte Carlo (QMC) simulations. For the Hubbard model we use the auxiliary-field approach and for Heisenberg systems the stochastic series expansion algorithm and concentrate on bipartite lattices where the QMC simulations are free of the negative sign problem. The hyperbolic lattices have an extensive number of sites on the boundary, such that one has to distinguish between bulk and total density of states (DOS). The considered lattices are characterized by a Dirac-like total DOS, Schläfli indices $\{p,q\}=\{10,3\}$ and $\{8,3\}$, as well as by flat bands, $\{8,8\}$. The Dirac total DOS cuts off the logarithmic divergence of the staggered spin susceptibility and allows for a finite $U$ metal-to-insulator transition. This transition has the same mean-field exponents as for the Gross-Neveu transition in Euclidean space. We argue that this transition is induced by the open-boundary conditions and that it will be absent in the periodic case. In the presence of flat bands we observe the onset of magnetic ordering at any finite $U$. This conclusion holds even though the bulk DOS is constant at the Fermi energy for the three considered lattices. The magnetic state at intermediate coupling can be described as a global antiferromagnet. It breaks the $C_p$ rotational and time-reversal symmetries but remains invariant under combined $C_p\mathcal{T}$ transformations. The state is characterized by macroscopic ferromagnetic moments, that globally cancel. We observe that fluctuations on the boundary of the system are greatly enhanced: While spin-wave calculations predict the breakdown of antiferromagnetism on the boundary but not in the bulk, QMC simulations show a marked reduction of the staggered moment on the edge of the system.

21 pages, 18 figures

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• Bose-Einstein condensation in exotic lattice geometries
• Classical spin liquids from frustrated Ising models in hyperbolic space